Asymptotic properties of derivatives of the Stieltjes polynomials

Let w λ (x): = (1 - x 2) λ - 1 / 2 and p λ,n (x) be the ultraspherical polynomials with respect to w λ (x). then,we denote the stieltjes polynomials with respect to w λ (x) by e λ,n + 1 (x) satisfying - 1 1 w λ (x) p λ,n (x) e λ,n + 1 (x) x m d x = 0,0 ≤ m n + 1,- 1 1 w λ (x) p λ,n (x) e λ,n + 1 (x) x m d x ≠ 0,m = n + 1. in this paper,we investigate asymptotic properties of derivatives of the stieltjes polynomials e λ,n + 1 (x) and the product e λ,n + 1 (x) p λ,n (x). especially,we estimate the even-order derivative values of e λ,n + 1 (x) and e λ,n + 1 (x) p λ,n (x) at the zeros of e λ,n + 1 (x) and the product e λ,n + 1 (x) p λ,n (x),respectively. moreover,we estimate asymptotic representations for the odd derivatives values of e λ,n + 1 (x) and e λ,n + 1 (x) p λ,n (x) at the zeros of e λ,n + 1 (x) and e λ,n + 1 (x) p λ,n (x) on a closed subset of (- 1,1),respectively. these estimates will play important roles in investigating convergence and divergence of the higher-order hermite-fejér interpolation polynomials. © 2012 hee sun jung and ryozi sakai.